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Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems
Ground state homoclinic solutions for a second-order Hamiltonian system
Department of Mathematics, Xiangnan University, Chenzhou, Hunan 42300, China |
$ \ddot{u}-L(t)u+\nabla W(t, u) = 0, $ |
$ t\in {\mathbb{R}}, u\in {\mathbb{R}}^{N} $ |
$ L: \mathbb{R}\rightarrow {\mathbb{R}}^{N\times N} $ |
$ W: {\mathbb{R}}\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}} $ |
$ L $ |
$ W $ |
$ t $ |
$ 0 $ |
$ \sigma\left(-\frac{d^2}{dt^2} +L\right) $ |
References:
[1] |
A. Ambrosetti and V. Coti Zelati,
Multiple homoclinic orbits for a class of conservative systems, Rend. Semmin. Mat. Univ. Padova, 89 (1993), 177-194.
|
[2] |
F. Antonacci and P. Magrone,
Second order nonautonomous systems with symmetric potential changing sign, Rend. Mat. Appl., 18 (1998), 367-379.
|
[3] |
P. Caldiroli and P. Montechiari,
Homoclinic orbites for second order Hamiltonian systems with potential changing sign, J. Commu. Appl. Nonlinear Anal., 1 (1994), 97-129.
|
[4] |
S. T. Chen and X. H. Tang,
Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin Dyn Syst-A, 38 (2018), 2333-2348.
doi: 10.3934/dcds.2018096. |
[5] |
V. Coti Zelati, I. Ekeland and E. Sere,
A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.
doi: 10.1007/BF01444526. |
[6] |
V. Coti Zelati and P. H. Rabinowitz,
Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.
doi: 10.1090/S0894-0347-1991-1119200-3. |
[7] |
Y. H. Ding and M. Girardi,
Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dynam. Systems Appl., 2 (1993), 131-145.
|
[8] |
Y. H. Ding,
Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113.
doi: 10.1016/0362-546X(94)00229-B. |
[9] |
Y. H. Ding and C. Lee,
Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413.
doi: 10.1016/j.na.2008.10.116. |
[10] |
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
![]() |
[11] |
Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996.
doi: 10.1007/978-3-0348-9029-8. |
[12] |
W. Kryszewski and A. Szulkin,
Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472.
|
[13] |
G. B. Li and A. Szulkin,
An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
doi: 10.1142/S0219199702000853. |
[14] |
Y. Lv and C. L. Tang,
Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198.
doi: 10.1016/j.na.2006.08.043. |
[15] |
M. Izydorek and J. Janczewska,
Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential Equations, 219 (2003), 375-389.
doi: 10.1016/j.jde.2005.06.029. |
[16] |
W. Omana and M. Willem,
Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
|
[17] |
Z. Q. Ou and C. L. Tang, Existence of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213. Google Scholar |
[18] |
A. Pankov,
Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
doi: 10.1007/s00032-005-0047-8. |
[19] |
P. H. Rabinowitz,
Periodic and Heteroclinic orbits for a periodic Hamiltonian systems, Ann.
Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 331-346.
doi: 10.1016/S0294-1449(16)30314-6. |
[20] |
P. H. Rabinowitz,
Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
doi: 10.1017/S0308210500024240. |
[21] |
P. H. Rabinowitz and K. Tanaka,
Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.
doi: 10.1007/BF02571356. |
[22] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[23] |
X. H. Tang,
Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on $ \mathbb{R} ^N$, Canad. Math. Bull., 58 (2015), 651-663.
doi: 10.4153/CMB-2015-019-2. |
[24] |
X. H. Tang,
Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.
doi: 10.1007/s11425-014-4957-1. |
[25] |
X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15. https://doi.org/10.1007/s10884-018-9662-2. Google Scholar |
[26] |
X. H. Tang and L. Xiao,
Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 351 (2009), 586-594.
doi: 10.1016/j.jmaa.2008.10.038. |
[27] |
X. H. Tang and L. Xiao,
Homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 71 (2009), 1140-1152.
doi: 10.1016/j.na.2008.11.038. |
[28] |
X. H. Tang and X. Y. Lin,
Homoclinic solutions for a class of second-order Hamiltonian systems, J. Math. Anal. Appl., 354 (2009), 539-549.
doi: 10.1016/j.jmaa.2008.12.052. |
[29] |
X. H. Tang and X. Y. Lin,
Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Royal Soc. Edinburgh A, 141 (2011), 1103-1119.
doi: 10.1017/S0308210509001346. |
[30] |
J. Wang, F. Zhang and J. Xu,
Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 366 (2010), 569-581.
doi: 10.1016/j.jmaa.2010.01.060. |
[31] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[32] |
D. L. Wu, X. P. Wu and C. L. Tang,
Homoclinic solutions for a class of nonperiodic and noneven second-order Hamiltonian systems, J. Math. Anal. Appl., 367 (2010), 154-166.
doi: 10.1016/j.jmaa.2009.12.046. |
[33] |
D. L. Wu, X. P. Wu and C. L. Tang,
Subharmonic and homoclinic solutions for second order Hamiltonian systems with new superquadratic conditions, Chaos Solitons and Fractals, 73 (2015), 183-190.
doi: 10.1016/j.chaos.2015.01.019. |
[34] |
X. Wu and W. Zhang,
Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398.
doi: 10.1016/j.na.2011.03.059. |
[35] |
J. Yang and F. Zhang,
Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials, Nonlinear Anal.RWA, 10 (2009), 1417-1423.
doi: 10.1016/j.nonrwa.2008.01.013. |
[36] |
M. H. Yang and Z. Q. Han,
The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials, Nonlinear Anal.RWA, 12 (2011), 2742-2751.
doi: 10.1016/j.nonrwa.2011.03.019. |
[37] |
Q. Q. Zhang,
Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Commun. Pure Appl. Anal., 14 (2015), 1929-1940.
doi: 10.3934/cpaa.2015.14.1929. |
[38] |
Q. Q. Zhang,
Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163.
doi: 10.1090/S0002-9939-2015-12107-7. |
[39] |
Z. Zhou and J. S. Yu,
On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212.
doi: 10.1016/j.jde.2010.03.010. |
[40] |
W. M. Zou,
Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287.
doi: 10.1016/S0893-9659(03)90130-3. |
show all references
References:
[1] |
A. Ambrosetti and V. Coti Zelati,
Multiple homoclinic orbits for a class of conservative systems, Rend. Semmin. Mat. Univ. Padova, 89 (1993), 177-194.
|
[2] |
F. Antonacci and P. Magrone,
Second order nonautonomous systems with symmetric potential changing sign, Rend. Mat. Appl., 18 (1998), 367-379.
|
[3] |
P. Caldiroli and P. Montechiari,
Homoclinic orbites for second order Hamiltonian systems with potential changing sign, J. Commu. Appl. Nonlinear Anal., 1 (1994), 97-129.
|
[4] |
S. T. Chen and X. H. Tang,
Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin Dyn Syst-A, 38 (2018), 2333-2348.
doi: 10.3934/dcds.2018096. |
[5] |
V. Coti Zelati, I. Ekeland and E. Sere,
A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.
doi: 10.1007/BF01444526. |
[6] |
V. Coti Zelati and P. H. Rabinowitz,
Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.
doi: 10.1090/S0894-0347-1991-1119200-3. |
[7] |
Y. H. Ding and M. Girardi,
Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dynam. Systems Appl., 2 (1993), 131-145.
|
[8] |
Y. H. Ding,
Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113.
doi: 10.1016/0362-546X(94)00229-B. |
[9] |
Y. H. Ding and C. Lee,
Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413.
doi: 10.1016/j.na.2008.10.116. |
[10] |
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
![]() |
[11] |
Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996.
doi: 10.1007/978-3-0348-9029-8. |
[12] |
W. Kryszewski and A. Szulkin,
Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472.
|
[13] |
G. B. Li and A. Szulkin,
An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
doi: 10.1142/S0219199702000853. |
[14] |
Y. Lv and C. L. Tang,
Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198.
doi: 10.1016/j.na.2006.08.043. |
[15] |
M. Izydorek and J. Janczewska,
Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential Equations, 219 (2003), 375-389.
doi: 10.1016/j.jde.2005.06.029. |
[16] |
W. Omana and M. Willem,
Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
|
[17] |
Z. Q. Ou and C. L. Tang, Existence of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213. Google Scholar |
[18] |
A. Pankov,
Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
doi: 10.1007/s00032-005-0047-8. |
[19] |
P. H. Rabinowitz,
Periodic and Heteroclinic orbits for a periodic Hamiltonian systems, Ann.
Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 331-346.
doi: 10.1016/S0294-1449(16)30314-6. |
[20] |
P. H. Rabinowitz,
Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
doi: 10.1017/S0308210500024240. |
[21] |
P. H. Rabinowitz and K. Tanaka,
Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.
doi: 10.1007/BF02571356. |
[22] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[23] |
X. H. Tang,
Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on $ \mathbb{R} ^N$, Canad. Math. Bull., 58 (2015), 651-663.
doi: 10.4153/CMB-2015-019-2. |
[24] |
X. H. Tang,
Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.
doi: 10.1007/s11425-014-4957-1. |
[25] |
X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15. https://doi.org/10.1007/s10884-018-9662-2. Google Scholar |
[26] |
X. H. Tang and L. Xiao,
Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 351 (2009), 586-594.
doi: 10.1016/j.jmaa.2008.10.038. |
[27] |
X. H. Tang and L. Xiao,
Homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 71 (2009), 1140-1152.
doi: 10.1016/j.na.2008.11.038. |
[28] |
X. H. Tang and X. Y. Lin,
Homoclinic solutions for a class of second-order Hamiltonian systems, J. Math. Anal. Appl., 354 (2009), 539-549.
doi: 10.1016/j.jmaa.2008.12.052. |
[29] |
X. H. Tang and X. Y. Lin,
Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Royal Soc. Edinburgh A, 141 (2011), 1103-1119.
doi: 10.1017/S0308210509001346. |
[30] |
J. Wang, F. Zhang and J. Xu,
Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 366 (2010), 569-581.
doi: 10.1016/j.jmaa.2010.01.060. |
[31] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[32] |
D. L. Wu, X. P. Wu and C. L. Tang,
Homoclinic solutions for a class of nonperiodic and noneven second-order Hamiltonian systems, J. Math. Anal. Appl., 367 (2010), 154-166.
doi: 10.1016/j.jmaa.2009.12.046. |
[33] |
D. L. Wu, X. P. Wu and C. L. Tang,
Subharmonic and homoclinic solutions for second order Hamiltonian systems with new superquadratic conditions, Chaos Solitons and Fractals, 73 (2015), 183-190.
doi: 10.1016/j.chaos.2015.01.019. |
[34] |
X. Wu and W. Zhang,
Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398.
doi: 10.1016/j.na.2011.03.059. |
[35] |
J. Yang and F. Zhang,
Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials, Nonlinear Anal.RWA, 10 (2009), 1417-1423.
doi: 10.1016/j.nonrwa.2008.01.013. |
[36] |
M. H. Yang and Z. Q. Han,
The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials, Nonlinear Anal.RWA, 12 (2011), 2742-2751.
doi: 10.1016/j.nonrwa.2011.03.019. |
[37] |
Q. Q. Zhang,
Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Commun. Pure Appl. Anal., 14 (2015), 1929-1940.
doi: 10.3934/cpaa.2015.14.1929. |
[38] |
Q. Q. Zhang,
Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163.
doi: 10.1090/S0002-9939-2015-12107-7. |
[39] |
Z. Zhou and J. S. Yu,
On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212.
doi: 10.1016/j.jde.2010.03.010. |
[40] |
W. M. Zou,
Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287.
doi: 10.1016/S0893-9659(03)90130-3. |
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